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Use of relaxation and correlation methods for NLTE problems

A. Sapar and R. Poolamäe

 

1. General introduction

One of main problems in the physics of stellar atmospheres is to model the deviations from the local thermodynamical equilibrium and to compute corresponding stellar spectra. Usually this problem has been studied by the use of different mathematical iteration methods which do not have clear physical meaning. As a rule, the methods are very bulky ones and convergence is slow.

To remove some of these drawbacks we propose to replace the equations of statistical equilibrium for the populations of atomic states and corrections to the corresponding equation of radiative transfer by the time-dependent equations and to solve them as the evolutionary equations from some adopted initial values. Such method can be named the relaxation method, it simulates corresponding physical process and leads to self-consistent equilibrium values of NLTE populations of atomic states.

To reduce the number of equations, the concept of strongly correlated or clustered states which are with each other in the thermal equilibrium is treated shortly.

 

2. Equations of relaxing evolution

Let us put down now the formulae ruling the evolutionary relaxation process. General formula for a population change rates of states $i$ can be written in the following compact form

$$\dot n_i=\sum_ k (P_{ki}n_k-P_{ik}n_i)=-P_i n_i+\sum_ k P_{ki}n_k,~~~~P_i=\sum_ k P_{ik}.$$ (1)

The transition probabilities, or more correctly the transition rates, $P_{ki}$ from upper state $k$ to lower state $i$ include contributions by spontaneous electron transitions $A_{ki}$, induced electron transitions $A_{ki}N_{ki}$ (where $N_{ki}$ is the population of the photon phase-space cell corresponding to electron transition considered) and the collision transitions $C_{ki}$, i.e.

$$P_{ki}=A_{ki}(1+N_{ki})+C_{ki}, ~~~~k>i$$ (2)

and for transitions in the opposite direction

$$P_{ik}=A_{ik}N_{ik}+C_{ik}, ~~~~i<k.$$ (3)

In these formulae $N_{ki}=N_{ik}$ and the statistical weights of the final states have been incorporated in the expressions of transition probabilities. The system of equations is independent for each chemical element $Z$ and for it holds the particle number conservation law and thus

$$\sum \limits_{i\in Z} n_i=n_Z=A_Z n,~~~\sum \limits_{i\in Z... ..._e=\sum_Z \sum_{i\in Z}Z_i n_i~~~n=\sum_Z \sum_{i\in Z} n_i.$$ (4)

In these equations $A_Z$ is relative abundance of element $Z$ by number and $Z_i$ is the ionization multiplicity particles in state $i$. The system of equations for the atomic state population relaxing evolution can be easily generalized for the case if also molecules exist in stellar atmosphere. In the result the system of equations turns into the non-linear relative to atomic state populations, but as we show further this circumstance does not complicate or modify the method of solution of these equations.

Let us specify now in more detail the expressions for transition probabilities. First we consider the bound-bound transitions. If the cross-section for spontaneous electron transition $k\rightarrow i$ is $\sigma_{ki}(\nu)$ then

$$A_{ki}=\int \alpha_{ki}d\nu, ~~~\alpha_{ki}=8\pi{\sigma_{ki} \over \lambda^2}$$ (5)

and for the light-induced transitions

$$A_{ki}N_{ki}=\int \alpha_{ki}N_\nu d\nu.$$ (6)

Further, the transition rate due to electron collisions

$$C_{ki}=n_e\int c _{ ki}v f(v)dv$$ (7)

where $c_{ ki}$ is the collision cross-section and $f(v)$ is the Maxwellian electron velocity distribution

$$f(v)dv={4 \over \sqrt{\pi}} e^{-v^2/v_T^2} {v^2 dv \over v_T^3}, ~~~~~v_T^2={2\kappa T \over m}$$ (8)

or

$$vf(v)dv={2v_T \over \sqrt{\pi}}e^{-x}xdx, ~~~~x={v^2 \over v_T^2}.$$ (9)

For the collision rates from $n_i^\star C_{ik}=n_k^\star C_{ki}$ it follows that

$$g_i e^{-E_i/\kappa T}C_{ik}=g_ke^{-E_k/\kappa T}C_{ki}$$ (10)

where $E_k$ is the excitation energy of the state $k$.

The formulae reduce finding of transition rates to the finding of adequate cross-sections $\sigma_{ki}$ and $c_{ ki}$.

As the next problem we specify the probabilities for bound-free and free-bound transitions. First, we take into account that the cross-sections of photo-ionization $\sigma_{ic}(\nu)$ and of photo-recombination $\varsigma_{ci}(\nu)$ are connected by the Milne formula

$$\varsigma_{ci}=\biggl({ h\nu\over cmv }\biggr )^2{g_i \over g_c}\sigma_{ic}, ~~~~h\nu={mv^2 \over 2}+E_{ci}.$$ (11)

The thermal ionization ratio $r^\star_{ci}$ is given by the Saha equation which for state population ratios can be used in the form

$$r^\star_{ci}={n_c^\star\over n_i^\star}={S_c \over g_i}e^{... ... \over n_e} \biggl ({2\pi m\kappa T \over h^2}\biggr )^{3/2}$$ (12)

where $g_c$ is the statistical weight for the ground state of the ion species characterized by subscript $c$. Further, the recombination probability

$$A_{ci}=n_e\int \limits_0^\infty \varsigma_{ci}vf(v)dv.$$ (13)

Using the Milne equation and going to integration over frequencies starting from the threshold frequency $\nu_c$, we obtain

$$A_{ci}=r^\star_{ci} \int \limits_{\nu_c}^\infty \alpha_{ic... ...a T} d\nu, ~~~\alpha_{ic}=8\pi {\sigma_{ic}\over \lambda^2}.$$ (14)

The contribution by the light-induced recombinations

$$A_{ci}N_{ci}=r^\star_{ci}\int \limits_{\nu_c}^\infty \alpha_{ic}e^{-h\nu /\kappa T}N_\nu d\nu.$$ (15)

For the processes of photo-ionization

$$A_{ic}N_{ic}=\int \limits_{\nu_c}^\infty \alpha_{ic}N_\nu d\nu.$$ (16)

For the thermal equilibrium case, if $k=c$ in (  ) , i.e. if the ionization and recombination transitions are considered, holds

$$n_c^\star P_{ci}=n_i^\star P_{ic}.$$ (17)

These formulae give a simple form for including the processes of photo-ionization and photo-recombination into the transition rate equations. The free-free transitions do not give deviations from the LTE and thus they do not give any contribution to the transition matrices.

The expression for the ionization and recombination rates by electron collisions is given by

$$C_{ic}=n_e \int c_{ic}v f(v) dv.$$ (18)

For the collision rates from $n_c^\star C_{ci}=n_i^\star C_{ic}$ it follows that

$$S_c e^{-E_c/\kappa T}C_{ci}=g_ie^{-E_i/\kappa T}C_{ic}.$$ (19)

 

3. Equation of radiative transfer

The equation of radiative transfer for plane-parallel stellar atmospheres can be written for the photon state occupation numbers $N_\nu$ in the form

$${\partial N_\nu \over c\partial t}+\mu{dN_\nu \over dx}= -\... ... _\nu N_\nu+\varepsilon_\nu(1+N_\nu)-\alpha^e(N_\nu-S^e_\nu),$$ (20)

where

$$\alpha_\nu=\sum_i\alpha_\nu^i,~~~ \varepsilon_\nu=\sum_i\varepsilon_\nu^i.$$ (21)

Here the transition indices $i$ incorporate bound-bound, bound-free and free-free transitions and by $e$ is denoted light scattering on free electrons.

Defining the monochromatic optical depth for absorption $\tau_\nu$ by

$$d\tau_\nu=(\alpha_\nu-\varepsilon_\nu) dx$$ (22)

we can write the equation of radiative transfer in the form

$${\partial N_\nu \over c\partial t}+\mu{dN_\nu \over d\tau_\... ...~~ S_\nu={\varepsilon_\nu \over \alpha_\nu-\varepsilon_\nu}.$$ (23)

For the bound-bound transitions

$$\alpha^i_\nu= n_{l_i}\sigma^i_\nu, ~~~~~~ \varepsilon^i_\nu=n_{u_i}\varsigma^i_\nu.$$ (24)

Here $n_{u_i}$ and $n_{l_i}$ are respectively the population for the upper and lower atomic state of given transition $i$. The cross-sections of absorption $\sigma^i_\nu$ and of emission $\varsigma^i_\nu$ are connected by $n_{l_i}^\star\alpha^i_\nu N^\star_i=n_{u_i}^\star\epsilon^i_\nu (1+N^\star_i)$ from where we obtain

$$\alpha^{jc}_\nu= n_j \sigma^{jc}_\nu,~~~~~ \varepsilon^{cj}_\nu=n_c\varsigma^{cj}_\nu,~~~~$$ (25)

$$g_{l_i}\sigma^i_\nu=g_{u_i}\varsigma^i_\nu.$$ (26)

Similarly, for the photo-ionization and photo-recombination processes

$$\alpha^{jc}_\nu= n_j \sigma^{jc}_\nu,~~~~~ \varepsilon^{cj}_\nu=n_c\varsigma^{cj}_\nu,~~~~$$ (27)

where $\sigma^{jc}_\nu$ and $\varsigma^{cj}_\nu$ are connected by

$$n_j^\star \sigma^{jc}_\nu N^\star_\nu= n_c^\star\varsigma^{cj}_\nu(1+N^\star_\nu)$$ (28)

from where

$$n^\star_{c}\varsigma^{cj}_\nu=n^\star_j\sigma_\nu^{jc} e^{-h\nu/\kappa T}.$$ (29)

For the free-free transitions which are always thermalized

$$\alpha^c_\nu=n_c\sigma^c_\nu,~~~\varepsilon^c_\nu= e^{-h\nu/\kappa T}\alpha^c_\nu.$$ (30)

In the case of LTE we obtain

$$~~~S_\nu=S_\nu^\star={1 \over e^{h\nu/\kappa T}-1}.$$ (31)

For the light scattering processes on free electrons

$$\alpha^e=n_e\sigma_e,~~~~\varepsilon^e_\nu=n_e\sigma_e S^e... ...nu,\nu^\prime,\theta)d\mu^\prime d\varphi^\prime d\nu^\prime$$ (32)

where $\phi$ is the frequency redistribution function depending on the angle of scattering $\theta$. Now we take into account that

$$\sigma^i_\nu=\sigma_0^i \phi^i_\nu,~~~\varsigma^i_\nu=\varsigma_0^i \phi^i_\nu$$ (33)

where $\phi^i_\nu$ is the normalized line profile function and the quantities $\sigma_0^i$,$\varsigma_0^i$ are the corresponding cross-sections in the line center of transition $i$.

The formalism used by us takes into account that only the scattering on free electrons is a real second-order process in terms of Feynman diagrams. Only this process cannot be reduced to independent processes of absorption and emission.

As the next step we find the expressions needed for computation of quantities $N_i$.

The total monochromatic optical depth $t_\nu$ and the total emissivity $\epsilon_\nu$ in radiative transfer are given by

$$t_\nu=\int a_\nu dx,~~~~a_\nu=\alpha_\nu+\alpha^e,~~~\epsilon_\nu=\varepsilon_\nu+\varepsilon_\nu^e.$$ (34)

Using these quantities we can write for given transition $ik$

$$N_{ik}=\int K_{ik} dx, ~~~~K_{ik}=\int_0^\infty\epsilon_\nu E_1(\vert\Delta t_\nu\vert)\phi_{ik} d\nu.$$ (35)

As the next step we find the expressions needed for computation of quantities $\dot N_{ik}$.

$$\dot\alpha_\nu=\sum_i (\dot n_{l_i}\sigma_\nu^i-\dot n_{u_... ...ma^{i}_\nu+\dot n_e\sigma_e,~~\dot t_\nu=\int \dot a_\nu dx.$$ (36)

Using these quantities we can write

$$\dot N_{ik}=\int \dot K_{ik} dx, ~~~~\dot K_{ik}=\int_0^\in... ...psilon \dot E_1(\vert\Delta t_\nu\vert)\bigr )\phi_{ik} d\nu$$ (37)

where

$$\dot E_1(\vert\Delta t_\nu\vert)=e^{-\vert\Delta t_\nu\vert}{ \Delta\dot t_\nu \over \Delta t_\nu}.$$ (38)

 

4. Temporal relaxation computations

The formulae enable to follow relaxed evolution choosing adequately the values of time steps. The formulae needed are

$$d\ln n_i={\dot n_i \over n_i}dt, ~~~~ n_i(dt)=n_i(0) e^{d\ln n_i}$$ (39)

and

$$d\ln N_{ik}={\dot N_{ik} \over N_{ik}}dt, ~~~~~ N_{ik}(dt)=N_{ik}(0) e^{d\ln N_{ik}}.$$ (40)

We have adopted the formulae of time evolution in the exponential form, because this corresponds in the best way to exponential relaxation processes of tending to equilibrium state and avoids the negative populations which can appear if the time step $dt$ is taken too long.

 

5. Correlation method

If the difference of excitation energies of two or more quantum states $j$ and $j^\prime$ is small and they are connected by electric dipole transitions then holds

$$A_{j^\prime j}(1+N_{j^\prime j})<<C_{j^\prime j}.$$ (41)

In this case the states are with each other in the thermodynamical equilibrium. This circumstance of strong correlation between the states allows to treat them as a single cluster-state or super-state. Using the notation $l$ for the cluster-state and making use of cluster-state partition function

$$g_l=\sum\limits_{j\in l}g_je^{-(E_j-E_l)/\kappa T},~~~~ E_l=min(E_j,j\in l)$$ (42)

of the cluster-state population

$$n_l=\sum \limits_{j\in l}n_j$$ (43)

and taking into account that in cluster-states holds

$${ n_j\over n_l}= { n^\ast_j\over n^\ast_l}$$ (44)

we get for corresponding transition probability expression

$$P_l=\sum\limits_{j\in l} {P_j n^\ast_j\over n^\ast_l}.$$ (45)

The transition probability to the cluster-state is

$$P_{kl}=\sum\limits_{j\in l} P_{kj}.$$ (46)

These operations reduce the original system of equations to considerably smaller system clustering the high-excitation states.

 

6. Contribution by molecules

For the diatomic molecules instead of the Saha equation we have the dissociation equation which is characterized by the dissociation coefficient $K(T)$. This equation for diatomic molecules is

$${n_A n_B \over n_{AB}}=K_{A,B}(T)$$ (47)

where

$$K_{A,B}(T)={S_d } e^{-E_d/\kappa T}(1-e^{-E_v/\kappa T }),... ...AB}4\pi I} \biggl ({2\pi M\kappa T \over h^2}\biggr )^{1/2}.$$ (48)

Quantities $g_A$, $g_B$ and $g_{AB}$ are correspondingly the statistical weights of ground states of atoms $A$ and $B$ and for molecules ${AB}$ where $E_d$ is the dissociation energy and $E_v$ is the vibrational energy of the molecule ground state. Further, $I$ is the maximal inertial moment and $M$ the reduced mass of molecule considered, i.e.

$$M={M_AM_B\over M_A+M_B}.$$ (49)

For three-atomic molecules hold similar equations of thermal equilibrium and then for three possible formation ways of molecules $ABC$ hold corresponding different equations

$${n_{AB} n_C \over n_{ABC}}=K_{AB,C}(T),~~ {n_{AC} n_B \over... ...{ABC}}=K_{AC,B}(T),~~ {n_{BC} n_A \over n_{ABC}}=K_{BC,A}(T).$$ (50)

However, the expressions for these dissociation constants are more complicated.

For three-atomic molecules we can also write the equation of statistical equilibrium in the form

$${n_A n_B n_C \over n_{ABC}}=K_{ABC}(T).$$ (51)

Assuming that the populations of molecules are in statistical equilibrium with the populations of the corresponding atoms in the ground state, we can treat them as components of the cluster states. This means that we can replace the atomic ground state populations $n_A$ by

$$n_A^\prime=f_A n_A,~~~f_A=1+\sum_B{n_B\over K_{AB}}+\sum_{B,C}{n_Bn_C\over K_{ABC}}+...$$ (52)

treating $f_A$ as an additional multiplier to the statistical weight of the ground state. In this expression also the atoms in molecules can coincide. Thus, if the molecules are taken into account then in the equations (  ) the ground-state statistical weight $g_A$ must be replaced by

$$g_A^\prime=f_A g_A.$$ (53)

 

7. Contribution by Strømgren ? states

The high-excitation or Strømgren states with partition function contribution $u_h$ are clustered with adjacent ion ground state. These states are almost free and the electrons on far orbits generate but marginal shifts of energy levels. The total partition function $u=u_l+u_h$. For the cluster state and for the lower state total population we obtain correspondingly

$$n^\prime={u_l \over u}n~~~,n_c^\prime=n_c+{u_h \over u_l}n.$$ (54)

Replacing these expressions in the Saha equation we obtain

$${n^\prime_c \over n^\prime}={u_h\over u_l}+{u_c \over u_l}S_c e^{-E_c/\kappa T}$$ (55)

 

8. Relaxational temperature change

Integrating the equation of radiative transfer expressed for energy over $\mu$ and $\nu$ we can write for the accumulation rate of internal energy density

$${\dot E}=\int (\alpha_\nu \bar N_\nu-\varepsilon_\nu S_\nu^\star) {2h\nu \over \lambda^2} d\nu.$$ (56)

As the internal energy we take into account the kinetic energy of particles ans the summary energy of dissociation, ionization and excitation corresponding in average to particles in state $n_j$. Thus

$$E={3\over 2}nkT+\sum_{j} E_j n_j$$ (57)

and

$$\dot E={3\over 2}(\dot n_e kT+nk\dot T)+\sum_{j} E_j \dot n_j.$$ (58)

From this expression we get the rate of temperature change $\dot T$ which must be also taken into account in the case of correction of model atmosphere due to NLTE effects. In the cases if also change of the convective energy transport must be taken into account appear additional terms which we do not study in the present paper.

 

9. Solution of the equation of radiative transfer

The only terms which are to be taken into account in the equation of radiative transfer are the Rayleigh and Thompson scattering on electrons. In the case of Rayleigh scattering on bound electrons with profile function we can assume that the redistribution corresponds only to the thermal motion of scattering atomic particles.

The other mechanism of real scattering is the Thompson scattering on free electrons which gives essential thermal broadening in frequencies of spectral lines. In very hot stars due to dilution of Planck radiation distribution it can also generate presence of emission lines in observed stellar spectra.

For solution of the equation of radiative transfer we propose to use a variant of the lambda iteration. In this scheme we use as argument the optical depth for absorption processes. Ignoring in the starting approximation the scattering source term such choice gives for the source function the Planck expression. Using iteratively the resulting radiation intensity we include the corresponding order scattering processes. In consequent iteration cycles the source term corresponding to absorption is not the Planck expression due to changed state populations. Due to circumstance that multiplicity of real scattering is small even the usual iterative seems to be good enough to be applied.